Optimal. Leaf size=50 \[ A b^2 x+\frac {1}{3} b (b B+2 A c) x^3+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{7} B c^2 x^7 \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1598, 380}
\begin {gather*} A b^2 x+\frac {1}{5} c x^5 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{7} B c^2 x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^4} \, dx &=\int \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2+b (b B+2 A c) x^2+c (2 b B+A c) x^4+B c^2 x^6\right ) \, dx\\ &=A b^2 x+\frac {1}{3} b (b B+2 A c) x^3+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{7} B c^2 x^7\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} A b^2 x+\frac {1}{3} b (b B+2 A c) x^3+\frac {1}{5} c (2 b B+A c) x^5+\frac {1}{7} B c^2 x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 49, normalized size = 0.98
method | result | size |
default | \(\frac {B \,c^{2} x^{7}}{7}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{5}}{5}+\frac {\left (2 A b c +b^{2} B \right ) x^{3}}{3}+A \,b^{2} x\) | \(49\) |
risch | \(\frac {1}{7} B \,c^{2} x^{7}+\frac {1}{5} x^{5} A \,c^{2}+\frac {2}{5} x^{5} b B c +\frac {2}{3} x^{3} A b c +\frac {1}{3} b^{2} B \,x^{3}+A \,b^{2} x\) | \(51\) |
gosper | \(\frac {x \left (15 B \,c^{2} x^{6}+21 A \,c^{2} x^{4}+42 x^{4} b B c +70 A b c \,x^{2}+35 b^{2} B \,x^{2}+105 b^{2} A \right )}{105}\) | \(54\) |
norman | \(\frac {\left (\frac {1}{5} A \,c^{2}+\frac {2}{5} b B c \right ) x^{8}+\left (\frac {2}{3} A b c +\frac {1}{3} b^{2} B \right ) x^{6}+A \,b^{2} x^{4}+\frac {B \,c^{2} x^{10}}{7}}{x^{3}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.50, size = 48, normalized size = 0.96 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{5} \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 53, normalized size = 1.06 \begin {gather*} A b^{2} x + \frac {B c^{2} x^{7}}{7} + x^{5} \left (\frac {A c^{2}}{5} + \frac {2 B b c}{5}\right ) + x^{3} \cdot \left (\frac {2 A b c}{3} + \frac {B b^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.88, size = 50, normalized size = 1.00 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {2}{5} \, B b c x^{5} + \frac {1}{5} \, A c^{2} x^{5} + \frac {1}{3} \, B b^{2} x^{3} + \frac {2}{3} \, A b c x^{3} + A b^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 48, normalized size = 0.96 \begin {gather*} x^3\,\left (\frac {B\,b^2}{3}+\frac {2\,A\,c\,b}{3}\right )+x^5\,\left (\frac {A\,c^2}{5}+\frac {2\,B\,b\,c}{5}\right )+\frac {B\,c^2\,x^7}{7}+A\,b^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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